Theorem lejoin1 18102

Description: A join's first argument is less than or equal to the join. (Contributed by NM, 16-Sep-2011.)

Hypotheses

Ref	Expression
joinval2.b	⊢ 𝐵 = (Base‘𝐾)
joinval2.l	⊢ ≤ = (le‘𝐾)
joinval2.j	⊢ ∨ = (join‘𝐾)
joinval2.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
joinval2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
joinval2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
joinlem.e	⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∨ )

Assertion

Ref	Expression
lejoin1	⊢ (𝜑 → 𝑋 ≤ (𝑋 ∨ 𝑌))

Proof of Theorem lejoin1

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		joinval2.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		joinval2.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		joinval2.j	. . 3 ⊢ ∨ = (join‘𝐾)
4		joinval2.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
5		joinval2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		joinval2.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7		joinlem.e	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∨ )
8	1, 2, 3, 4, 5, 6, 7	joinlem 18101	. 2 ⊢ (𝜑 → ((𝑋 ≤ (𝑋 ∨ 𝑌) ∧ 𝑌 ≤ (𝑋 ∨ 𝑌)) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → (𝑋 ∨ 𝑌) ≤ 𝑧)))
9	8	simplld 765	1 ⊢ (𝜑 → 𝑋 ≤ (𝑋 ∨ 𝑌))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ⟨cop 4567   class class class wbr 5074  dom cdm 5589  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  lecple 16969  joincjn 18029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-lub 18064  df-join 18066

This theorem is referenced by:  joinle  18104  latlej1  18166